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Weights with Maximal Symmetry and Failures of the MacWilliams Identities

Jay A. Wood

TL;DR

This work addresses when MacWilliams-type duality holds for generalized $w$-weight enumerators on finite rings with maximal symmetry, revealing that duality largely fails beyond a few narrow settings. It develops an orbit-multiplicity framework, introducing the $W$-matrix (and its averaged version $W_0$) to relate multiplicity data to orbit-weights, and constructs counterexamples via chain rings and matrix rings by matching $w$-weight enumerators while forcing different dual enumerators through singleton contributions. Key positive results occur for the Hamming weight over finite fields and a special chain-ring case ($q=2$, $m=2$) as well as the homogeneous weight on $M_{2 imes 2}( ext{F}_2)$; in general, the homogeneous weight on larger matrix rings or chain rings with $m>2$ does not satisfy the MacWilliams identities. The rank-partition enumerator framework and Kravchuk transforms further illuminate duality behavior in matrix rings, with explicit constructions and degeneracy analyses provided. Collectively, the results delineate the boundaries of duality-respecting generalized weight enumerators in non-field algebraic settings and quantify when duality-based transforms can be expected to fail.

Abstract

This paper examines the $w$-weight enumerators of weights $w$ with maximal symmetry over finite chain rings and matrix rings over finite fields. In many cases, including the homogeneous weight, the MacWilliams identities for $w$-weight enumerators fail because there exist two linear codes with the same $w$-weight enumerator whose dual codes have different $w$-weight enumerators.

Weights with Maximal Symmetry and Failures of the MacWilliams Identities

TL;DR

This work addresses when MacWilliams-type duality holds for generalized -weight enumerators on finite rings with maximal symmetry, revealing that duality largely fails beyond a few narrow settings. It develops an orbit-multiplicity framework, introducing the -matrix (and its averaged version ) to relate multiplicity data to orbit-weights, and constructs counterexamples via chain rings and matrix rings by matching -weight enumerators while forcing different dual enumerators through singleton contributions. Key positive results occur for the Hamming weight over finite fields and a special chain-ring case (, ) as well as the homogeneous weight on ; in general, the homogeneous weight on larger matrix rings or chain rings with does not satisfy the MacWilliams identities. The rank-partition enumerator framework and Kravchuk transforms further illuminate duality behavior in matrix rings, with explicit constructions and degeneracy analyses provided. Collectively, the results delineate the boundaries of duality-respecting generalized weight enumerators in non-field algebraic settings and quantify when duality-based transforms can be expected to fail.

Abstract

This paper examines the -weight enumerators of weights with maximal symmetry over finite chain rings and matrix rings over finite fields. In many cases, including the homogeneous weight, the MacWilliams identities for -weight enumerators fail because there exist two linear codes with the same -weight enumerator whose dual codes have different -weight enumerators.
Paper Structure (26 sections, 86 theorems, 261 equations, 1 figure)

This paper contains 26 sections, 86 theorems, 261 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generalities
  3. Preliminaries
  4. MacWilliams identities
  5. Linear codes via multiplicity functions
  6. Singletons in dual codes
  7. Finite Chain Rings
  8. Definitions and a positive result
  9. Modules over a chain ring
  10. Two families of linear codes with the same wwe
  11. Analysis of dual codewords of low weight
  12. Weak monotonicity and final arguments
  13. Symmetrized enumerators and examples
  14. Matrix Rings over Finite Fields
  15. Matrix modules, their orbits, and a positive result
  16. ...and 11 more sections

Key Result

Lemma 2.3

Suppose $R$ is Frobenius and $C \subseteq R^n$ is an additive code. Then

Figures (1)

  • Figure 1: Relations among enumerators

Theorems & Definitions (196)

  • Lemma 2.3
  • Remark 2.4
  • Example 2.6
  • Example 2.7
  • Example 2.9
  • Theorem 3.2: MacWilliams identities MR2939359MR0149978
  • Remark 3.3
  • Corollary 3.4
  • Remark 3.10
  • Definition 3.12
  • ...and 186 more